Introduction

\section{Introduction}

• In the past session, you have encountered a vast array of financial models
  • Basic ARIMA models for the mean equation
  • GARCH extensions to deal with heteroscedasticity
  • Multivariate GARCH models that deal with dependence modeling
• Theoretical problem arises when we talk about dependence
  • Capturing co-movement between financial asset returns with linear correlation has been the staple approach in modern finance since the birth of Harry Markowitz’s portfolio theory
  • But linear correlation is only appropriate when the dependence structure (or joint distribution) follow a normal distribution
• Enter copulas - flexible framework to model general multivariate dependence

GIF

References

• The work on copulas are vast, as this isn’t just a technique but a field of statistics
  • Go to reference: Joe (2014)
  • Financial applications: Ruppert (2011), Carmona (2014)
  • Actuarial: Charpentier (2014)
• For applications on South Africa is limited, but there are some works for emerging markets

Goal

• To introduce to you an extension in the field of risk management to a multivariate space with multiple assets.
• Grasp basic concepts and generators within the field of copulas
  • Learn to walk, before we can run
  • Revisit your statistics
• Understand the field of copulas to such an extent that you might go on to do a PhD in this field ;-)

Fields where copulas are applied

• Quantitative finance
  • Stress-tests and robustness checks
  • “Downside/crisis/panic regimes” where extreme downside events are important
  • Pool of asset evaluation
  • Latest development: Vine Copulas
  • Hot research page here
• Civil engineering
• Warranty data analysis
• Turbulent combustion
• Medicine

\section{Understanding Copulas}

Introduction to copulas

• Copula stems from the latin verb copulare; bond or tie.
  • Regulatory institutions are under pressure to build robust internal models to account for risk exposure
  • Fundamental ideology of these internal models rely on joint dependency among whole basket of mixed instruments
  • This issue can be addresed through the copula instrument
  • It functions as a linking mechanism between uniform marginals such as the cdf
• Copula theory was first developed by Sklar in 1959 Nelsen (2007).

Introduction to copulas (Sklar)

• Sklar’s theorem forms the basis for copula models as:
  • It does not require identical marginal distributions and allows for n-dimensional expansion
• Let \(X\) be a random variable with marginal cumulative distribution function:
  • \(F_X(x) = \mathcal{P}(X \leq x)\)
  • Probability that random variable \(X\) takes on a value less or equal to point of evaluation
  • \(F_X(x) \sim U(0,1)\)

Introduction to copulas (Sklar cont.)

• If we now denote the inverse CDF (Quantile funtion) as \(F_x^{-1}\)
  • \(U \sim U(0,1)\) then \(F_x^{-1}(U) \sim F(X)\)
• This allows a simple way for us to simulate observations from the \(F_X\) provided the inverse cdf, \(F_X^{-1}\) is easy to calculate
• Think, median is \(F_X^{-1}(0.5)\)

Lets have a look visually

Definitions and basic properties

• Define the uniform distribution on an interval \((0,1)\) by \(U(0,1)\), i.e the probability of a random variable \(U\) satisfying \(P(U \leq u) = u\) for \(u \in (0,1)\)
  • This is the quantile function \((Q = F^{-1})\) Probability transformation implies that if \(X\) has a distribution function \(F\), then \(F(X) \sim U(0,1)\) iff \(F\) is continous

Definitions and basic properties (cont)

Definition (Copula): A d-dimensional copula is the distrubutiom function \(\mathcal{C}\) of a random vector \(U\) whose components \(U_k\) are uniformly distributed \begin{align} C(u_1, \ldots, u_d) = P(U_1 \leq u_1, \ldots, U_d \leq u_d), (u_1, \ldots, u_d) \in (0,1)^d \end{align}\vspace{-1.5cm} Thus Sklar’s theorem states: \begin{align} C(F_1(x_1), \ldots,F_d(x_d)) &= P(U_1 \leq F_1(x_1), \ldots, U_d \leq F_d(x_d))\notag\\ &= P(F_1^{-1}(U_1) \leq x_1, \ldots, F_d^{-1}(U_d) \leq x_d)\notag\\ &= F(x_1, \ldots, x_d) \end{align}

Joint distribution function:

• This represents the joint distribution function function \(F\) can be expressed in terms of a copula \(C\) and the marginal distristributions \((F_1, \ldots, F_d)\). Modeling them seperately

• Easy Def: A Copula is a function that couples the joint distribution function to its univariate marginal distribution

• Dependence or correlation coefficient dependent on marginal distributions. This one to one mapping of correlation and dependece only works in case of elliptical joint distribution

• For copulas, we use Kendall’s Tau - non-linear concordance measure

Kendall’s Tau

• Let \((X_1, Y_1)\) and \((X_2, Y_2)\) be i.i.d random vectors, each with joint distribution function \(H\)

• Tau is then defined as the probability of concordance minus the probability of discordance \begin{align} \tau = \tau_{X,Y} = P((X_1 - X_2)(Y_1 - Y_2) > 0) - P((X_1 - X_2) - (Y_1 - Y_2) < 0) \end{align}\vspace{-1.5cm}

• Tau is the difference between the probability that the observed data are in the same order versus the probability that the observed data are no in the same order

\section{Copulas}

Meeting the generator

The Gaussian copula \(C_p^{Ga}\) is a distribution over the unit cube \([0,1]^{d}\). This represents a linear correlation matrix \(\rho\). \(C_p^{Ga}\) is defined as the distribution function of a random vector \((\psi(X_1),\ldots, \psi(X_d))\), where \(\psi\) is the univariate standard normal distribution function where \(X \sim, N_d(0,\rho)\) \begin{align} C_p^{Ga}=\Phi _{R}\left(\Phi ^{-1}(u_{1}),\dots ,\Phi ^{-1}(u_{d})\right) \end{align}\vspace{-1.5cm}

where \(\Phi^{-1}\) is the inverse cumulative distribution function of a standard normal and \(\Phi_{R}\) is the joint cumulative distribution

Student T Copula

The student’s t-copula \(C_{\upsilon,\rho}^t\) of \(d\)-dimension can be characterised by parameter \(\upsilon\geq0\) degrees of freedom and linear correlation matrix \(\rho\). The random vector \(X\) has a \(t^d(0,\rho\upsilon)\) distribution with univariate function \begin{align} C_{\upsilon,\rho}^t &= \mathcal{P}(t_\upsilon(X_1)\leq(u_1), \ldots, t_\upsilon(X_d)\leq u_d)\\ &=t_{\upsilon,\rho}^d(t_{\upsilon}^{-1}(u_1),\ldots, t_\upsilon^{-1}(u_d)) \end{align}

Archimedean Copulas

• Most common Archimedean copulas admit an explicit formula (Guassian dont)
  • Archimedean copulas are popular because they allow modeling dependence high dimensions
  • Does this with only one only one parameter, governing the strength of dependence.

Illustration

See Joe (2014)

Vine-Copulas

• A vine is a graphical tool for labeling constraints in high-dimensional probability distributions
• Regular Vines from part of what is know as pair copula construction
• Trees are constructed between copulas based on what is know as maximum spanning degree (Or concordance measure)

Under suitable differentiability conditions, any multivariate density \(F_{1\ldots n}\) on \(n\) variables may be represented in closed form as a product of univariate densities and (conditional) copula densities on any R-vine \(V\)

Vine copulas kurowicka2006

The R-vine copula density is uniquely identified according to Theorom 4.2 of @kurowicka2006: \begin{align} c(F_1(x_1),\cdots, F_d(x_d)) = \prod_{i=1}^{d-1}\prod_{e\in E_i}c_{j(e),k(e)|D(e)}\left(F(x_{j(e)}|\mathbf{x}_{D(e)})\right) \label{r_vine_eq} \end{align}

• Introduction to VineCopula
• Website for the research here

Introduction of Copula-MGARCH model

References